NUCLEAR AND PARTICLE PHYSICS -3

- November 05, 2021

Electric moment of nuclei:

Atomic nucleus is a positively charged body of finite dimensional. The potential φ (r, θ) due to azimuthally symmetric distribution of charges can be expanded in ascending power of 1/r where r is the distance of the point from the origin of the co-ordinate nucleus.

Where 𝑃_𝑛’s are the Legendre polynomial of different order.

The first term in the expansion corresponds to potential due to an electric monopole which is point charge +Ze, Z being the atomic number.

The second term in the expansion corresponds to the potential due to an electric dipole which turns out to be zero. Since the protons are distributed through out the volume, the electric dipole moment (p=Zed) of the nucleus in its ground state vanishes. This hold good for all odd orders( i.e. octupole moment) which are all zero for nucleus.

The third term in the expansion corresponds to the potential due to the electric quadrupole moment Q of the cylindrically symmetric charge distribution. The four charge quadrupole has net charge and electric dipole moment zero.

Atomic nuclei, the spin of which is ≥1 have quadrupole moment other than zero. The quadrupole moment has a + sign if the nucleus is extended as a spin axis and minus sign if the nucleus is extended in a plane perpendicular to the spin axis.

One can generate quadrupole moment by displacement of electric diploe with its sign ‘p’ reversed. The quadrupole moment is given by Q= 2𝑝𝑑^′ where 𝑑^′ is the displacement. Q=

2𝑞𝑑𝑑^′ where q is the charge, d𝑑^′ has the dimension of area (m² ) which taken as the unit of q.

The intrinsic quadrupole moment of a nucleus is

𝑄

where the integration is carried out over the whole volume of

the nucleus. 𝑟^′(𝑥^′, 𝑦^′, 𝑧′) is measured from the centre of mass of the nucleus. Normally the

expression is divided by e, so that it’s unit is length ^2.

Isotopic spin:

The charge symmetry is charge independent of nuclear forces and almost equality of masses of the neutron and proton suggest that the neutron and proton are the same particles in two different charge states. We introduce mathematically their dimensional ‘charge space’ which is isotopic spin space or isospin space. The neutrons and protons are described by an isotopic spiner field in the isospace.

A spin 1/2 particle can exist in two spin sates. In absence of any interaction, the two spin states having the component 𝑠_𝑧= + 1/2 𝑜𝑟 − 1/2 have same energy. However, in presence of magnetic field, the symmetry is broken which have different energies

Neutron and proton together is given the common name nucleon.

A nucleon is a particle with isospin 𝐼 = 1/2 𝜏 where 𝜏 is an operator.

The z component is 𝐼₃= ± 1/2 , 𝐼₃= (1/2)𝜏₃= ± 1/2